Quantitative Reasoning Final Exam (QTN-561)

Your Results: The correct answer for each question is indicated by a

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Choose the best answer 1 INCORRECT

The difference between a random variable and a probability distribution is A) A random variable does not include the probability of an event. B) A random variable can only assume whole numbers. C) A probability distribution can only assume whole numbers. D) None of the above. Feedback: The correct answer is a. A random variable is a value resulting from an experiment, while a probability distribution is a listing of all possible outcomes and their associated probability.

2 INCORRECT

Which of the following is not a requirement of a binomial distribution? A) A constant probability of success. B) Only two possible outcomes. C) A fixed number of trails. D) Equally likely outcomes. Feedback: The correct answer is d. A binomial distribution has only two possible outcomes on each trial, results from counting successes over a series of trials, the probability of success stays the same from trial to trial and successive trials are independent.

3 CORRECT

The mean and the variance are equal in A) All probability distributions. B) The binomial distribution. C) The Poisson distribution. D) The hypergeometric distribution.

4 INCORRECT

In which of the following distributions is the probability of a success usually small? A) Binomial B) Poisson C) Hypergeometric D) All distribution Feedback: The correct answer is b. That's why it is often referred to as the “law of improbable events.”

5 INCORRECT

Which of the following is not a requirement of a probability distribution? A) Equally likely probability of a success. B) Sum of the possible outcomes is 1.00.

C) The outcomes are mutually exclusive. D) The probability of each outcome is between 0 and 1. Feedback: The correct answer is a. Only the classical notion of probability requires the events to be equally likely. 6 INCORRECT

For a binomial distribution A) n must assume a number between 1 and 20 or 25. B) π must be a multiple of .10. C) There must be at least 3 possible outcomes. D) None of the above. Feedback: The correct answer is d. A binomial distribution has only two possible outcomes on each trial, results from counting successes over a series of trials, the probability of success stays the same from trial to trial and successive trials are independent.

7 INCORRECT

Which of the following is a major difference between the binomial and the hypergeometric distributions? A) The sum of the outcomes can be greater than 1 for the hypergeometric. The probability of a success changes from trial to trial in the hypergeometric B) distribution. C) The number of trials changes in the hypergeometric distribution. D) The outcomes cannot be whole numbers in the hypergeometric distribution. Feedback: The correct answer is b. A typical case where the hypergeometric distribution applies is sampling without replacement. Hence the probability of a success changes from trial to trial.

8 INCORRECT

In a continuous probability distribution A) Only certain outcomes are possible. B) All the values within a certain range are possible. C) The sum of the outcomes is greater than 1.00 D) None of the above. Feedback: The correct answer is b. Continuous implies without interruption. So it includes all numbers, without exception, in a range.

9 INCORRECT

For a binomial distribution with n = 15 as π changes from .50 toward .05 the distribution will A) Become more positively skewed. B) Become more negatively skewed C) Become symmetrical. D) All of the above. Feedback: The correct answer is a. As the likelihood of success gets smaller, the positive tail gets relatively longer.

10 INCORRECT

The expected value of the a probability distribution A) Is the same as the random variable. B) Is another term for the mean. C) Is also called the variance. D) Cannot be greater than 1. Feedback: The correct answer is b. The average or mean describes what you “expect.”

Your Results: The correct answer for each question is indicated by a

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Choose the best answer 1 INCORRECT

A sample A) Is a part of the population. B) Has more than 30 observations. C) Is usually identified as N. D) All of the above. Feedback: The correct answer is a. That's its definition.

2 INCORRECT

Which of the following is not a reason for sampling? A) The destructive nature of certain tests. B) The physical impossibility of checking all the items in the population. C) The adequacy of sample results. D) All of the above are reasons for sampling. Feedback: The correct answer is d. They are all correct.

3 INCORRECT

Which of the following is not a method of probability sampling? A) Random sampling B) Systematic sampling C) Stratified sampling D) All of the above are methods of probability sampling. Feedback: The correct answer is d. They are all probability sampling methods.

4 INCORRECT

In a simple random sample th A) Every k item is selected to be in the sample. B) Every item has a chance to be in the sample. C) Every item has the same chance to be in the sample. D) All of the above.

Feedback: The correct answer is c. Answer a describes systematic sampling and answer b describes sampling, but not a simple random sample. 5 CORRECT

Suppose a population consisted of 20 items. How many different samples of n = 3 are possible? A) 6840 B) 1140 C) 20 D) 120

6 INCORRECT

The difference between the sample mean and the population mean is called the A) Population mean. B) Population standard deviation. C) Standard error of the mean. D) Sampling error. Feedback: The correct answer is d. That's its definition.

7 INCORRECT

The mean of the all the sample means and the population mean will A) Always be equal. B) Always be normally distributed. C) Characterized by the standard error of the mean. D) None of the above. Feedback: The correct answer is a. It's a fact!

8 INCORRECT

Suppose we have a population that follows the normal distribution. Which of the following statements is correct regarding the distribution of sample means? A) The population standard deviation is always unknown. B) The distribution of samples means will follow the uniform distribution. C) The distribution of the sample means will also follow the normal distribution. D) None of the above is correct. Feedback: The correct answer is c. The process of averaging the sample values doesn't change the normality of the distribution. In fact because of the Central Limit theorem effect it reinforces it.

9 INCORRECT

Suppose we have a population that does not follow the normal distribution. If we select sample of what size will the distribution approximate the normal distribution? A) 2 B) 5 C) 20 D) 30 Feedback: The correct answer is d. That's enough for the Central Limit theorem to take

effect. 10 INCORRECT

The standard error of the mean is A) The standard deviation of the sampling distribution of sample means. B) Always normally distributed. C) Sometimes less than 0. D) None of the above.

Feedback: The correct answer is a. That's its definition.

Your Results: The correct answer for each question is indicated by a A)Always an estimate of the population mean. B Always equal to the population value. )

. 1INCORRECT A point estimate is

C)An estimate of the population parameter. D)None of the above. Feedback: The correct answer is C. You could be estimating a standard deviation as well as a mean and you're never sure it's exactly the same as the population parameter. See pages 360 and 361. 2INCORRECT A confidence interval A)Always includes the population parameter. B Decreases in width as the sample size is increased. ) C)Cannot include a value of 0. D)None of the above. Feedback: The correct answer is B. A confidence interval can include any value and you can only hope it includes the population parameter. See page 372 for an explanation of why B is the correct answer. Section 9.2 (pages 364-366) gives a general introduction to the concept of confidence intervals. 3INCORRECT If we wish to decrease the width of a confidence interval, we would not do which of the following: A)Increase the size of the sample. B Reduce the size of the population. ) C)Decrease the level of confidence. D)None of the above. Feedback: The correct answer is D. The width would increase if you reduced the size of the sample. Page 372 contains an explanation of the factors that determine interval width. 4INCORRECT We wish to develop a confidence interval for the population mean. The shape of the population is not known, but we have a sample of 40 observations. We decide to use the 92 percent level of confidence. What is the appropriate value of z? A)1.9 6 B 1.6 ) 5 C)2.5

8 D)1.7 5 Feedback: The correct answer is D. Remember to look for an area of 0.9200/2 or 0.4600 in the body of the standard normal (z) table. See pages 366 –367, and try the SELF-REVIEW 9-2 on page 367. 5INCORRECT Which of the following statements is not a characteristic of the t distribution? A)It is a continuous distribution. B It has a mean of 0. ) C)It is symmetrical. D)Like z, there is only one t distribution. Feedback: The correct answer is D. There are different t distributions depending on the degrees of freedom. See page 374 for a description of the characteristics of the t distribution. 6CORRECT We wish to develop a confidence interval for the population mean. The population follows the normal distribution, we know the population standard deviation, and have a sample of 10 observations. We decide to use the 90 percent level of confidence. The appropriate value to represent the level of confidence is A)z = 1.65 B z = 1.96 ) C)t = 1.833 D)t = 1.812 E t = 2.262 ) Feedback: The correct answer is A. If the population follows the normal distribution, so will the sampling distribution of the mean. See the discussion about the correct choice when estimating µ (Z or t) at the beginning of section 9.4 on page 373, continued on pages 377-378. 7INCORRECT The fraction or ratio of a sample possessing a certain trait is called a A)Population. B Mean. ) C)Confidence interval. D)Proportion Feedback: The correct answer is D. That's its definition. See page 382. 8INCORRECT To develop a confidence interval for a proportion A)We need to meet the binomial conditions. B The sample should be at least 100. ) C)p should be less than .05. D)None of the above. Feedback: The correct answer is D. Option A is close, but we also need a large enough sample to let the Central Limit theorem work and approximate the binomial distribution by a normal. See pages 382-383. 9INCORRECT The finite population correction factor is used when A)n is more than 30. B N is more than 1,000. )

C)np is greater than 5. D)n/N is more than .05. Feedback: The correct answer is D. Then our sample (n) is a significant part of the population (N). See page 393. 10INCORRECT We wish to estimate the population proportion. We want to be 95 percent confident of our results and we want the estimate to be within .01 of the population parameter. No estimate of the population proportion is available. What value should we use for p? A)1.9 6 B 0.0 ) 1 C)0. 5 D)We cannot complete the problem; we need more information.

Your Results: The correct answer for each question is indicated by a . 1INCORRECT Which of the following is a correct statement about a probability? A)It may range from 0 to 1. B It may assume negative values. ) C)It may be greater than 1. D)It cannot be reported to more than 1 decimal place. E All the above are ) correct. Feedback: The correct answer is a. Zero represents the impossible and one, a sure thing. 2CORRECT An experiment is a A)Collection of events. B Collection of outcomes. ) C)Always greater than 1. D)The act of taking a measurement or the observation of some activity. E None of the above is correct. ) Feedback: The correct answer is d. That's its definition. 3INCORRECT Which of the following is not a type of probability? A)Subjective B Independent ) C)Relative frequency D)Classical

Feedback: The correct answer is b. "Independent" refers to a relation between events, not probabilities. 4CORRECT Events are independent if A)By virtue of one event occurring another cannot. B The probability of their occurrence is greater than ) 1. C)We can count the possible outcomes. D)The probability of one event happening does not affect the probability of another event happening. E None of the above. ) Feedback: The correct answer is d. That's its definition. 5INCORRECT The Special Rule of Addition is used to combine A)Independent events. B Mutually exclusive events ) C)Events that total more than one. D)Events based on subjective probabilities E Found by using joint probabilities. ) Feedback: The correct answer is b. There is no "double counting" among mutually exclusive two events. 6INCORRECT We use the General Rule of Multiplication to combine A)Events that are not independent. B Mutually exclusive events. ) C)Events that total more than 1.00. D)Events based on subjective probabilities E Found by using joint probabilities. ) Feedback: The correct answer is a. It considers dependent or conditional probability. 7INCORRECT When we find the probability of an event happening by subtracting the probability of the event not happening from 1, we are using A)Subjective probability B The complement rule. ) C)The general rule of addition. D)The special rule of multiplication E Joint probability ) Feedback: The correct answer is b. That's its definition. 8INCORRECT When we determine the number of combinations A)We are really computing a probability. B The order of the outcomes is not important. ) C)The order of the outcomes is important.

D)We multiple the likelihood of two independent trials. E None of the above. ) Feedback: The correct answer is b. Order is only important is permutations. 9INCORRECT Bayes' Theorem A)Is an example of subjective probability. B Can assume of value less than 0. ) C)Is used to revise a probability based on new or additional information. D)Is found by applying the complement rule. E None of the above. ) Feedback: The correct answer is c. That's a fact! 10INCORRECT The difference between a permutation and a combination is: A)In a permutation order is important and in a combination it is not. B In a permutation order is not important and in a combination it is important. ) C)A combination is based on the classical definition of probability. D)A permutation is based on the classical definition of probability. E None of the above. )

Results Reporter Out of 10 questions, you answered 3 correctly, for a final grade of 30%. 3 correct (30%) 7 incorrect (70%) 0 unanswered (0%) Choose the best answer

Your Results: The correct answer for each question is indicated by a

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1INCORRECT The science of statistics includes which of the following: A)Organizing data. B Presenting data. ) C)Interpreting data. D)All of the above. Feedback: All of these are correct answers. We also usually include collecting and analyzing data in our definition of statistics. 2CORRECT

In descriptive statistics our main objective is to A)Describe the population. B Describe the data we collected.

) C)Infer something about the population. D)Compute an average. Feedback: The best answer is b since we are trying to "summarize data in an informative way." 3CORRECT

Which of the following statements is true regarding a population? A)It must be a large number of values. B It must refer to people. ) C)It is a collection individuals, objects, or measurements. D)None of the above. Feedback: The correct answer is c. The population doesn't have to be large or refer only to people.

4INCORRECT Which of the following statements is true regarding a sample? A)It is a part of population. B It must contain at least five observations. ) C)It refers to descriptive statistics. D)All of the above are correct. Feedback: The correct answer is a, which is the definition of a sample. 5INCORRECT A qualitative variable A)Always refers to a sample. B Is not numeric. ) C)Has only two possible outcomes. D)All of the above are correct. Feedback: The correct answer is b. Examples of qualitative variables include hair color, religious affiliation, and gender. 6INCORRECT A discrete variable is A)An example of a qualitative variable. B Can assume only whole number values. ) C)Can assume only certain clearly separated values. D)Cannot be negative. Feedback: The correct answer is c. It could be fractions, such as ¼, ½, and ¾. 7CORRECT

A nominal scale variable is A)Usually the result of counting something.

B Has a meaningful zero point. ) C)May assume negative values. D)Cannot have more than two categories. Feedback: The best answer is a. Recall nominal variables can only be classified, but there is no natural order. 8INCORRECT The ratio scale of measurement A)Usually involves ranking. B Cannot assume negative values. ) C)Has a meaningful zero point. D)Is usually based on counting. Feedback: The correct answer is c. The ratio scale is the "highest" level of measurement. 9INCORRECT The ordinal scale of measurement A)Has a meaningful zero point. B Is based on ranks ) C)Cannot assume negative values. D)All of the above. Feedback: The correct answer is b. The word "order" is another way of saying rank. 1 Categories are exhaustive when INCORRECT 0 A)There is a meaningful zero point. B The objects can be ranked. ) C)Each object must appear in at least one category. D)Each object can be included in only one category. Feedback: The correct answer is c. That's the definition.

Your Results: The correct answer for each question is indicated by a

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Choose the best answer 1 INCORRECT

To conduct a nonparametric test the A) Population must follow the normal distribution. B) The standard deviation must be known. C) It is not necessary to make any assumption about the shape of the population. D) The data must be at least interval scale.

Feedback: The correct answer is c. The major feature of nonparametric tests is assumptions about the shape of the population are not necessary. This is different from earlier chapters when we assumed a normal population. 2 INCORRECT

Which of the following is not a characteristic of the X2 A) Its shape is based on the sample size. B) It is not negative. C) It is positively skewed. D) It approaches a normal distribution as the degrees of freedom increase. Feedback: The correct answer is a. The X2 distribution is based on the number of categories not the size of the sample. The other responses are all characteristics of the X2 distribution.

3 INCORRECT

In a goodness-of-fit test where the sample size is 200, there are 5 categories, and the significance level is .05. The critical value of X2 is A) 9.488 B) 11.070 C) 43.773 D) None of the above. Feedback: The correct answer is a. The degrees of freedom is the number of categories minus one, so df = 5 – 1 =4. Go a chi-square table and locate the column headed .05. Move down that column to the row with 4 degrees of freedom. The value is 9.488.

4 INCORRECT

In a goodness-of-fit test A) All the cell frequencies must be the same. B) There must be at least 30 observations. C) Forty percent of the cells must contain at least 10 observations. D) None of the above. Feedback: The correct answer is d. None of these statements are always correct.

5 CORRECT

In a contingency table A) The number of rows must be the same as the number of columns. B) A variable is classified according to two criteria. C) There must be at least 10 observations in each cell. D) All of the above.

6 INCORRECT

In a contingency table a sample of 400 people is classified by gender and hair color (4 groups: blond, brown, black, and red). How many degrees of freedom are there? A) 3 B) 8 C) 399 D) None of the above.

Feedback: The correct answer is a. The degrees of freedom is (r – 1)(c –1) = (2 – 1)(4 – 1) = 3. 7 INCORRECT

For a X2 goodness-of-fit test A) There is only one degree of freedom. B) The rejection region is in the upper right tail. C) The scale of measurement is interval. D) We must assume a normal population. Feedback: The correct answer is b. The data may be of nominal scale and the degrees of freedom are the number of categories minus one. For this X2 test all the rejection region is in the upper tail.

8 INCORRECT

To find the expected frequency in a contingency table A) Take the square root of the degrees of freedom. Multiple the row total by the column total and divide the result by the grand B) total. C) Use the total number of observations minus one. D) None of these. Feedback: The correct answer is b. To find the expected frequency is each cell we multiply the row total by the column total and divide by the grand total.

9 INCORRECT

Suppose we select a sample of 100 observations and organize them into 6 categories. We wish to investigate whether the number of observations could be the same in each of the categories in the population. How many degrees of freedom are there? A) 5 B) 97 C) 3 D) None of these. Feedback: The correct answer is a. The degrees of freedom is equal to the number of categories minus one, so df = 6 – 1 = 5.

10 INCORRECT

Under what conditions could the X2 distribution assume negative values? A) When the sample size is small. B) When the cell frequencies are all equal. C) When the degrees of freedom is 1. D) Never

Feedback: The correct answer is d. The X2 distribution cannot assume negative values because it is based on the squared difference between observed and expected frequencies. If the observed and expected values were the same, it would be 0.

Your Results:

The correct answer for each question is indicated by a . 1INCORRECT In a two-sample test of means for independent samples, the equal sign always appears in A)The null hypothesis. B The alternate hypothesis. ) C)The upper tail of the test statistic. D)None of the above. Feedback: The correct answer is a. The equality sign always is in the null hypothesis. The alternate hypothesis will never have an equal, it will be >,